Why do planets tend to rotate in the same direction although they have formed from tumbling asteroids?

You are right that the tilt of the asteroids are distributed in very random way, and that the rotation of the Solar nebula is a minor contributor to that tilt, and only skews it a little.

However, you are not right that randomness simply adds up. The randomness does in fact cancel out more and more when you combine a large amount of asteroids, until the rotation of the nebula becomes the dominant factor. This is related to the Law of large numbers.

For instance, throw a dice. The outcome is random. Throw 10 dices, calculate their sum, and divide by 10. Not so far from the average any more? You can do the same thing with thousands of dices, or millions of asteroids. When the number of asteroids that form an object is really high, the tilt is not going to be far from the average value, determined by the nebula's rotation.

The same argument goes for inclination, and the fact that even though the planets' orbits are elliptical, they are not as far from circular that a random orbit would be.